Ordered probability spaces

Let C be an open cone in a Banach space equipped with the Thompson metric with closure a normal cone. The main result gives sufficient conditions for Borel probability measures μ,ν on C with finite first moment for which μ≤ν in the stochastic order induced by the cone to be order approximated by sequences {μn}, {νn} of uniform finitely supported measures in the sense that μn≤νn for each n and μn→μ, νn→ν in the Wasserstein metric. This result is the crucial tool in developing a pathway for extending various inequalities on operator and matrix means, which include the harmonic, geometric, and arithmetic operator means on the cone of positive elements of a C⁎-algebra, to the space P1(C) of Borel measures of finite first moment on C. As an illustrative and important particular application, we obtain the monotonicity of the Karcher geometric mean on P1(A+) for the positive cone A+ of a C⁎-algebra A.